Question: Examples of simultaneous and quadratic equation with test
1. **Stating the problem:**
We will solve an example of simultaneous equations and a quadratic equation.
2. **Simultaneous equations example:**
Solve the system:
$$\begin{cases} 2x + 3y = 6 \\ x - y = 1 \end{cases}$$
3. **Formula and rules:**
Simultaneous equations are solved by substitution or elimination.
4. **Step-by-step solution for simultaneous equations:**
- From the second equation: $$x = y + 1$$
- Substitute into the first:
$$2(y + 1) + 3y = 6$$
- Simplify:
$$2y + 2 + 3y = 6$$
$$5y + 2 = 6$$
- Subtract 2:
$$5y = 4$$
- Divide both sides by 5:
$$y = \frac{\cancel{5}y}{\cancel{5}} = \frac{4}{5}$$
- Substitute back to find $$x$$:
$$x = \frac{4}{5} + 1 = \frac{4}{5} + \frac{5}{5} = \frac{9}{5}$$
5. **Quadratic equation example:**
Solve:
$$x^2 - 5x + 6 = 0$$
6. **Formula and rules:**
Use factorization or quadratic formula:
$$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$
where $$a=1$$, $$b=-5$$, $$c=6$$.
7. **Step-by-step solution for quadratic:**
- Calculate discriminant:
$$\Delta = (-5)^2 - 4 \times 1 \times 6 = 25 - 24 = 1$$
- Find roots:
$$x = \frac{-(-5) \pm \sqrt{1}}{2 \times 1} = \frac{5 \pm 1}{2}$$
- Roots:
$$x_1 = \frac{5 + 1}{2} = 3$$
$$x_2 = \frac{5 - 1}{2} = 2$$
**Final answers:**
- Simultaneous equations: $$x = \frac{9}{5}$$, $$y = \frac{4}{5}$$
- Quadratic equation: $$x = 3$$ or $$x = 2$$